Metric Geometry of Groups and Surfaces

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AbstractAward: DMS-0906086Principal Investigator: Moon DuchinThis award is funded under the American Recovery and Reinvestment Act of 2009(Public Law 111-5).The projects in this proposal are clustered around themes inlow-dimensional topology, asymptotic geometry, and the dynamicsof group actions. Teichmuller space and the mapping class groupare objects for special attention, and tools developed for thatsetting are often promising for wider study in metric geometryand geometric group theory. The PI proposes to build on recentwork on flat metrics, their length spectra, and the associatedboundary theory, as well as work on synthetic curvatureconditions and their applications to random walks. Specificavenues of future investigation include asymptotics on the spaceof singular flat surfaces; filling inequalities and higherdivergence functions adapted from symmetric spaces to grouptheory; and the use of intersection patterns of metrichalf-spaces to study the dynamics of isometries.While quite abstract, ideas in this work can also provebeautifully applicable-- for instance, Thurston's theory of flowson surfaces has direct and strong applications to the study ofmixing in fluid dynamics. More immediately, though, thisresearch area addresses foundational questions towardunderstanding geometric structure of complicated spaces in thelarge, in the small, and changing over time.